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I know how to interpret it and how to calculate it

But why does the sum of the products of the standardized values divided by n will always be between -1 and 1?

I would like to understand this mathematical property in an intuitive way, please.

I searched a lot for this answer but couldn't find an answer suitable for someone who is not a mathematician like me.

Thanks in advance.

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  • $\begingroup$ Because it is the cosine of some angle? Because of Cauchy-Schwarz? Because the division by the product of the square roots is a conspiracy to make the quotient come out to be between $\pm1$? This last is not totally false, and might appeal to a non mathematician. (It's just a sugar coated version of "because".) $\endgroup$ – kimchi lover Sep 8 '17 at 0:43
  • $\begingroup$ This is almost the same question as math.stackexchange.com/questions/158449/… but I'm holding off on calling this a "duplicate" for now because this one asks specifically for an intuitive explanation for a non-mathematician. $\endgroup$ – David K Sep 8 '17 at 0:50
  • $\begingroup$ Check out my answer here. The answer to your question is in there, and it is geometric. Covariance between two random variables is much like an ordinary dot product of two vectors. The variance of an individual random variable is like a vector being dotted with itself (i.e. square of its magnitude). Now the correlation coefficient formula looks exactly like cosine of the "angle between" the two random variables. I think this is an elegant interpretation. $\endgroup$ – jnez71 Sep 8 '17 at 3:57
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It's not clear to me if you are asking a purely technical question (to which the answer "it drops out of Cauchy Schwarz" is acceptable) or a motivational question (why would we expect it to be between $\pm1$, or how can we understand, in the context of applications, what the $\pm1$ bound "means").

So here is a stab at the 2nd kind of answer. In the context of least squares fitting of straight lines to data scatter plots, where you try to fit an affine function of $X$ to observed $Y$ values, one can ask for the fraction of variance in $Y$ "explained" by the affine function of $X$. The wikipedia article explains that this is the square of the correlation coefficient.

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